Math 5C 2011 Summer Session A Course Blog
Week 6
2011 July 29
Final exam.
2011 July 28
There will be a synopsis of the material from the course.
2011 July 27
We found the Laplacian in polar coordinates, solved another instance of the 2D wave equation, ran through a derivation of Bessel's
function inspired by the use of power series to solve ODEs, and discussed using orthonormal systems to solve PDEs.
2011 July 26
We solved an instance of the 2D wave equation. A
final review session is tomorrow, Wednesday July 27th, from 4pm to 7pm in
South
Hall 1607 (the Mathlab room).
2011 July 25
Solutions to Quiz 4, deriving the solution to an
instance of the heat equation, were provided at the end of class. At the
end of office hours, one student asked for one additional reference, in
addition to the alternative put on reserve in the library. Under fair use,
I've put up excerpts of Wylie's Advanced Engineering Mathematics
here.
Week 5
2011 July 22
On Thursday and Friday we went over PDEs and a
solution to the wave equation. I handed out a key and rubric to an old 5C
midterm, which included circulation, surface integral, and volumetric
computations. Office hours on Monday will be 11-12am and 4-5pm.
2011 July 21
Here
are some practice materials for the final and next weeks quiz.
2011 July 20
We thoroughly discussed 11.9. We started to talk
about partial differential equations. We were sidetracked by solving
coupled ODEs and relating the spectral theorem for matrices to Fourier
series.
2011 July 19
We considered absolute integrability and then
discussed Fourier integrals and various forms of the Fourier transform. We
discussed implementation of filters. We discussed the interplay between
multiplication and differentiation. We introduced convolution.
2011 July 18
Solutions
to Quiz 3. Bonus points will be provided to whomever finds mathematical errors in the
solutions first.
Week 4
2011 July 15
Project Day: We worked on problems. Our workflow
involved finding a Fourier series, making a table of Fourier coefficients
and successive approximations, calculating the L2 error using Parseval's
identity, and more.
2011 July 14
We discussed Parseval's identity and
L2-minimization in depth and included comments on projections of vector
spaces. We discussed ways in which one might choose the degree of a
truncation, ie the degree of the Fourier polynomial. We discussed Gibbs
phenomena.
2011 July 13
We used the double angle identity to prove the
orthogonal relations and went over the exercise suggested yesterday
[further details of Euler's formulas]. We worked on two examples which
will be similar to the upcomming quiz on Monday. We discussed even and odd
functions and how these correspond, respectively, to cosine and sine
series. We hinted at L2-minimization.
2011 July 12
The solutions to yesterday's quiz were reviewed. We
then focused on trigonometric series; briefly discussed convergence; and
noted for periodic, continuous functions with left and right derivatives
at all points that the trigonometric series can be computed using Euler's
formulas. We called our result the Fourier series. We discussed orthogonal
relations and inner products. The midterms were returned at the end of
class.
2011 July 11
There was a quiz, here
are the solutions.
After the quiz, we started talking about trigonometric series
and Fourier series at the end of class.
Week 3
2011 July 8
We further developed our notions of power series as
functions. We focused on the fact that the Taylor series associated to a
function and a center might acheive different values from the function
away from the center. On a related but different note, we discussed
differences the function and truncations of its Taylor series, aka Taylor
polynomials. We found a bound on error estimates which led us to a
remainder estimation theorem. The quiz on Monday will cover material from
our discussions on Tuesday, Thursday, and Friday. There will be an office
hour on Monday noon-1:30pm. Next week we will discuss Fourier Series.
2011 July 7
We spoke of power series as functions. We considered
differentiating and integrating a power series on the interior of its
interval of convergence. This brought us to a discussion of the Taylor
series. We found MacLaurin and Taylor series of familiar functions. We
will discuss error estimates and remainders tomorrow, along with a project
on Taylor series.
2011 July 6
Midterm. Solutions
2011 July 5
We discused power series and interval of convergence. We digressed and
discussed limsup and liminf. We discuss sections 10.7-10.10 according
to the following plan:
Power series
Interval of convergence, radius of convergence
Power series as functions and Representing
functions as power series
Differentiation/Integration of Power Series
Truncations of series
Taylor/MacLaurin Polynomials
Week 2
2011 July 1
Project day: we worked on problems related to the previous lecture,
briefly discussed evaluating series and then focused on techniques of
determining convergence. On Tuesday we will discuss power series and more,
but that material will not be on the exam on Wednesday. Remember the
office hours for next week will be Tuesday, July 5th, 4-5pm, and in South
Hall 6635.
2011 June 30
We concluded the formal discussion of convergence tests for series. In
total we
covered:
Checking sequences of partial sums
More webwork problems on series will be available shortly.
There will be a midterm review in SH6635 on Tuesday, July 5th, between
4-5pm. The midterm will cover the classical intergration theorems,
sequences, and series up to the convergence test for series. The midterm
will not cover Maclaurin nor Taylor polynomials/series, power series, etc.
Tomorrow we will work on a project related to series.
2011 June 29
We began our discussion of series which we thought of as sequences of
partial sums. We discussed the linearity of convergent series, geometric
series, telescoping series, and the divergence test. We will discuss
convergence tests tomorrow.
2011 June 28
We discussed sequences, at length. Our discussion included limit laws of
sequences, the linearity of limits, a useful limit law involving
continuous functions, the squeeze theorem, l'Hopital's rule, a heuristic
for growth of functions, etc. Tomorrow we will discuss series.
2011 June 27
We took a quiz
on the classical integration theorems. Here
are the solutions. We then started our focus on sequences and series. We spoke of convergence of sequences. Webwork
will be available shortly. Homework problems are from section 10.1, section 10.2, and be sure to read through section
10.3.
Week 1
2011 June 24
We discussed the theorem of Gauss in the context of
the fundamental theorem of calculs, the normal form of Green's theorem,
and the generalized Stoke's theorem. There will be a quiz on Monday
regarding the classical integration theorems. Next week we will concern
ourselves with series.
2011 June 23
We discussed the theorem of Stokes and went over examples. We parametrized
surfaces, computed normal vectors, calculated the curl of a vector field,
evaluated surface integrals, and determined when it's more appropriate to
calculate a circulation integral. Webwork is up and running.
2011 June 22
We discussed Green's theorem. It is a higher-dimensional
analogue of the fundamental theorem of calculus and a specific instance
of the Generalized Stokes' Theorem. We solved problems from
section 8.1. Office hours have been set to be Mondays 4-5pm and Fridays
11am-12pm.
2011 June 21
We met and discussed the course. We discussed the syllabus, which is
reproduced below. Let's discuss Green's theorem tomorrow. Read the textbook. Solve problems.
Course: Math 5C 2011 Summer Session A
Graduate Student Instructor: Rick Spjut
Office location: SH6432Q
Office hours will be set based on preferences indicated on Quiz
0.
Read your textbook.
Grading: The final is on July 29th and it is worth two fifths of your
grade.
There will be a midterm July 6th worth one fifth of your grade.
There will be four quizzes each Monday besides July 4th each worth two
twenty-fifths of your grade.
One twenty-fifth of your grade will come from completion of in-class
worksheets.
We will learn about Green's Theorem, Stoke's Theorem, series, Fourier
series, and PDEs.
UCSB math deparment's course description of 5C.